set "baseuri" "cic:/matita/nat/div_and_mod".
+include "datatypes/constructors.ma".
include "nat/minus.ma".
+
let rec mod_aux p m n: nat \def
match (leb m n) with
[ true \Rightarrow m
definition mod : nat \to nat \to nat \def
\lambda n,m.
match m with
-[O \Rightarrow m
+[O \Rightarrow n
| (S p) \Rightarrow mod_aux n n p].
interpretation "natural remainder" 'module x y =
apply div_aux_mod_aux.
qed.
+theorem eq_times_div_minus_mod:
+\forall a,b:nat. O \lt b \to
+(a /b)*b = a - (a \mod b).
+intros.
+rewrite > (div_mod a b) in \vdash (? ? ? (? % ?))
+[ apply (minus_plus_m_m (times (div a b) b) (mod a b))
+| assumption
+]
+qed.
+
inductive div_mod_spec (n,m,q,r:nat) : Prop \def
div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
rewrite > plus_minus.
rewrite > sym_times.
rewrite < H5.
-rewrite < sym_times.
+rewrite < sym_times.
apply plus_to_minus.
apply H3.
apply le_times_r.
rewrite < plus_n_O.rewrite < sym_times.reflexivity.
qed.
+lemma div_plus_times: \forall m,q,r:nat. r < m \to (q*m+r)/ m = q.
+intros.
+apply (div_mod_spec_to_eq (q*m+r) m ? ((q*m+r) \mod m) ? r)
+ [apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt ? r)
+ [apply le_O_n|assumption]
+ |apply div_mod_spec_intro[assumption|reflexivity]
+ ]
+qed.
+
+lemma mod_plus_times: \forall m,q,r:nat. r < m \to (q*m+r) \mod m = r.
+intros.
+apply (div_mod_spec_to_eq2 (q*m+r) m ((q*m+r)/ m) ((q*m+r) \mod m) q r)
+ [apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt ? r)
+ [apply le_O_n|assumption]
+ |apply div_mod_spec_intro[assumption|reflexivity]
+ ]
+qed.
(* some properties of div and mod *)
theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
intros.
-apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O).
-goal 15. (* ?11 is closed with the following tactics *)
-apply div_mod_spec_div_mod.
-unfold lt.apply le_S_S.apply le_O_n.
-apply div_mod_spec_times.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O);
+[2: apply div_mod_spec_div_mod.
+ unfold lt.apply le_S_S.apply le_O_n.
+| skip
+| apply div_mod_spec_times
+]
+qed.
+
+(*a simple variant of div_times theorem *)
+theorem lt_O_to_div_times: \forall a,b:nat. O \lt b \to
+a*b/b = a.
+intros.
+rewrite > sym_times.
+rewrite > (S_pred b H).
+apply div_times.
qed.
theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
assumption.reflexivity.
qed.
+theorem mod_SO: \forall n:nat. mod n (S O) = O.
+intro.
+apply sym_eq.
+apply le_n_O_to_eq.
+apply le_S_S_to_le.
+apply lt_mod_m_m.
+apply le_n.
+qed.
+
+theorem div_SO: \forall n:nat. div n (S O) = n.
+intro.
+rewrite > (div_mod ? (S O)) in \vdash (? ? ? %)
+ [rewrite > mod_SO.
+ rewrite < plus_n_O.
+ apply times_n_SO
+ |apply le_n
+ ]
+qed.
+
+theorem le_div: \forall n,m. O < n \to m/n \le m.
+intros.
+rewrite > (div_mod m n) in \vdash (? ? %)
+ [apply (trans_le ? (m/n*n))
+ [rewrite > times_n_SO in \vdash (? % ?).
+ apply le_times
+ [apply le_n|assumption]
+ |apply le_plus_n_r
+ ]
+ |assumption
+ ]
+qed.
+
(* injectivity *)
theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
change with (\forall n,p,q:nat.(S n)*p = (S n)*q \to p=q).
(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
+